Nearly tight frames and space-frequency analysis on compact manifolds

Abstract

Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let Δ be the Laplace–Beltrami operator on M. Say ${0 \neq f \in \mathcal{S}(\mathbb R^+)}$ , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ). Suppose f satisfies Daubechies’ criterion, and b  >  0. For each j, write M as a disjoint union of measurable sets E _j,k with diameter at most ba ^j, and measure comparable to ${(ba^j)^n}$ if ba ^j is sufficiently small. Take x _j,k ∈ E _j,k. We then show that the functions ${\phi_{j,k}(x)=\mu(E_{j,k})^{1/2} \overline{K_{a^j}}(x_{j,k},x)}$ form a frame for (I  −  P)L ²(M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I  −  P)L ² is in space and in frequency, we can describe which terms in the summation ${F \sim SF = \sum_j \sum_k \langle F,\phi_{j,k} \rangle \phi_{j,k}}$ are so small that they can be neglected. If n  =  2 and M is the torus or the sphere, and f (s)  =  se ^−s (the “Mexican hat” situation), we obtain two explicit approximate formulas for the φ _j,k, one to be used when t is large, and one to be used when t is small.